A man has three thousand euros for buying a car. He hesitates between two cars. He bought one of them with probability 1/2. One year later, his friend also wants to buy one of these two cars. He wants to take profit of the experience of his friend and asks his friend how many failures he had with his car. Modelling the number of failures with a Poisson distribution, could you help the friend for getting a new car?
More generally, the framework can be a joint distribution on $(X_0,X_1)$, and we focus on $(X_0 \wedge X_1,X_0 \vee X_1)$. Actually since the distributions of these two variables are stochastically dominated, the problem can be some kind of two-armed bandit problem where each arm delivers a random variable. If the two distributions are stochastically ordered, then the aim is to find the arm which corresponds to the distributiion which dominates the other one. The optimal function $\alpha$ corresponds to the cumulative function associated to probability measure which put most of its mass on the $x$ where $\PP(X_0 \vee X_1 \leq x)-\PP(X_0 \wedge X_1 \leq x)$ is maximal.